y-homeomorphism

The y-homeomorphism also dubbed crosscap slide, is an auto-homeomorphism (or self-homeomorphism) which can be defined only for non orientable surfaces whose genus is greater than one.

To define it we take a punctured Klein bottle $K_{0}=K\setminus{\rm int}\ D^{2}$ which can be consider as a pair of closed Möbius bands $M_{1},M_{2}$ , one sewed in the other by perforating with a disk (being disjoint from $\partial M_{1}$ ) and then identify the boundary of the second with the boundary of that disk, in symbols:

K_{0}=(M_{1}\setminus{\rm int}\ D^{2})\cup_{\partial}M_{2}

where $\partial=\partial D^{2}=\partial M_{2}$ . Other way to visualizing that, is by consider $K_{0}$ as the connected sum of ${\rm int}\ M_{1}$ with a projective plane ${\mathbb{R}}P^{2}$ .

Now, thinking that the removed disk $D^{2}$ was located with its center at some point in the core of $M_{1}$ , the second band, $M_{2}$ will have a pair of points on that part of the core in common with $\partial M_{2}$ .

So, the y-homeomorphism is defined by a isotopy leaving the boundary $\partial M_{1}$ fixed by sliding the second band $M_{2}$ one turn around the core of $M_{1}$ till the original position. The result is an automorphism of $K_{0}$ which maps $M_{2}$ into itself but reversing it.

To define this for genus greater than two just consider any other non orientable surface as a connected sum of a Kein bottle plus projective planes.

1.

D.R.J. Chillingworth. A finite set of generators for the homeotopy group of a non-orientable surface, Proc. Camb. Phil. Soc. 65(1969), 409-430.
2.

M. Korkmaz. Mapping Class Groups of Non-orientable Surfaces, Geometriae Dedicata 89 (2002), 109-133.
3.

W.B.R. Lickorish. Homeomorphisms of non-orientable two-manifolds, Math. Proc. Camb. Phil. Soc. 59 (1963), 307-317.

Title	y-homeomorphism
Canonical name	Yhomeomorphism
Date of creation	2013-03-22 15:42:26
Last modified on	2013-03-22 15:42:26
Owner	juanman (12619)
Last modified by	juanman (12619)
Numerical id	8
Author	juanman (12619)
Entry type	Definition
Classification	msc 54C10
Synonym	crosscap slide
Related topic	CrosscapSlide